So the degenerate part is taking a fixed sample of points no matter how big the spiral gets. The effect is like you're sampling at larger and larger scales, so you get to sample the structure of the spiral at larger and larger scales, and the diversity of patterns you see, is the visual representation of the spiral at these larger and larger scales through the lens of the sample you are taking. That's my explanation anyway.

in the spirit of hypnotic -- a while ago I was searching SO on how to animate canvas elements using js. I fell down a bit of a rabbit hole and a few hours later ended up with this:

https://codepen.io/867-5309/full/xxqGmVM

Reminds me of moiré patterns.

https://en.m.wikipedia.org/wiki/Moiré_pattern

This is found in magic angle graphene, which will likely yield a Nobel Prize in a few years: https://www.quantamagazine.org/when-magic-is-seen-in-twisted...

I wonder if there is a connection to these spirals and their segmentation lengths…

Reminds me of Junju Ito’s manga, Uzumaki, a masterful horror piece which involves a town becoming consumed with spirals, anyone else read/seen it?

>In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.

I'm surprised by that definition and always thought of a spiral as a curve with a monotonic signed-curvature function.

So, for example, the Euler/Cornu Spiral has a point of inflection where the curvature changes sign at the point of inflection, but the curvature increases continuously all the way from -infinity to + infinity as you travel along the length of the curve. So under my definition the whole Euler Spiral would count as a spiral, even though it stops revolving/emanating from a point just under 1/4 turn after the inflection point.

If you split a curve into segments at its curvature minimum and maximum points (vertices in the differential geometry sense [0]) then each segment has monotonic curvature and I'd define those as spiral segments. Vertices and monotonic curvature segments are preserved under inversion, which is mathematically useful.

In contrast, inflection points with zero curvature are not preserved under inversion. So the Euler spiral can be transformed by a suitable inversion to a curve like the one defined by Wikipedia, that is a curve emanating out from, for example, the origin.

Edit: just spotted this in the Wikipedia article on spirals 1]:

> Spirals which do not fit into this scheme of the first 5 examples:

> A Cornu spiral has two asymptotic points.

> The spiral of Theodorus is a polygon.

> The Fibonacci Spiral consists of a sequence of circle arcs.

> The involute of a circle looks like an Archimedean, but is not:

The Cornu spiral I've covered.

The spiral of Theodorus doesn't have a monotonic curvature function - it's a polygon approximation of the Archimedes Spiral, which does.

The Fibonacci Spiral's curvature function is a monotonic step-function.

The involute of a circle is a log-aesthetic curve, all of which have monotonic curvature functions. (The logarithmic spiral and the Euler spiral are also log-aesthetic curves.)

[0] https://en.wikipedia.org/wiki/Vertex_(curve)

[1] https://en.wikipedia.org/wiki/Spiral

This is a beautiful example of how to teach math and computer science (or simply, introducing students to "programming" or "coding") at the same time.

I find it interesting that the visualizations seem to cycle polygon complexity, from circles to squares to triangles and back again.

If you like this, you might also like: Belousov-Zhabotinsky reactions

https://www.youtube.com/watch?v=PpyKSRo8Iec

https://www.youtube.com/watch?v=kw9wF-GNjqs

They can form very neat patterns.

Reminds me of Geiss winamp visualisations. Recently installed ProjectM on my Android TV device. Mesmerising.

After watching the animated spiral at top of the page for maybe 45 seconds, I scrolled to read the text -- and experienced an interesting aftereffect: alternate lines of text seemed to be moving in opposite directions. Trippy!

Ooh, and I got an optical illusion for free. After staring at the spiral for a bit the text then seemed to want to spin as well.

Missed opportunity to title this article: “The Twisted World of Degenerate Spirals.”

Really wonderful.

Anyone who knows how to implement this with projectM, or GLSL?

Another example. http://gpu.pizza/#love

Tangential, but I am curious. What is the embedded animation/player made of?

Is this effect a version of signal aliasing?